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Strong convergence of parabolic rate $1$ of discretisations of stochastic Allen-Cahn-type equations

Máté Gerencsér, Harprit Singh

TL;DR

The paper studies fully discrete explicit schemes for 1+1D stochastic Allen–Cahn-type SPDEs driven by space–time white noise, and shows that while pointwise strong convergence is limited to a rate near 1/2, measuring the error in negative Besov norms yields almost sure convergence at rate essentially 1 (up to ε) via the Da Prato–Debussche framework. The authors develop a Besov-space analytic apparatus with discrete and continuous Besov norms, heat-kernel and error bounds, and a priori estimates to control the nonlinear discretisation error; a Grönwall argument then yields the main rate theorem. A complementary lower-bound result shows the rate cannot be improved in a distributional sense, highlighting a regularisation-by-noise effect in Besov norms. The work combines precise kernel estimates, discrete–continuous norm equivalences, and a careful a priori analysis to reveal improved convergence behavior for SPDE discretisations in a distributional setting, with concrete implications for numerical treatment of singular SPDEs.

Abstract

Consider the approximation of stochastic Allen-Cahn-type equations (i.e. $1+1$-dimensional space-time white noise-driven stochastic PDEs with polynomial nonlinearities $F$ such that $F(\pm \infty)=\mp \infty$) by a fully discrete space-time explicit finite difference scheme. The consensus in literature, supported by rigorous lower bounds, is that strong convergence rate $1/2$ with respect to the parabolic grid meshsize is expected to be optimal. We show that one can reach almost sure convergence rate $1$ (and no better) when measuring the error in appropriate negative Besov norms, by temporarily `pretending' that the SPDE is singular.

Strong convergence of parabolic rate $1$ of discretisations of stochastic Allen-Cahn-type equations

TL;DR

The paper studies fully discrete explicit schemes for 1+1D stochastic Allen–Cahn-type SPDEs driven by space–time white noise, and shows that while pointwise strong convergence is limited to a rate near 1/2, measuring the error in negative Besov norms yields almost sure convergence at rate essentially 1 (up to ε) via the Da Prato–Debussche framework. The authors develop a Besov-space analytic apparatus with discrete and continuous Besov norms, heat-kernel and error bounds, and a priori estimates to control the nonlinear discretisation error; a Grönwall argument then yields the main rate theorem. A complementary lower-bound result shows the rate cannot be improved in a distributional sense, highlighting a regularisation-by-noise effect in Besov norms. The work combines precise kernel estimates, discrete–continuous norm equivalences, and a careful a priori analysis to reveal improved convergence behavior for SPDE discretisations in a distributional setting, with concrete implications for numerical treatment of singular SPDEs.

Abstract

Consider the approximation of stochastic Allen-Cahn-type equations (i.e. -dimensional space-time white noise-driven stochastic PDEs with polynomial nonlinearities such that ) by a fully discrete space-time explicit finite difference scheme. The consensus in literature, supported by rigorous lower bounds, is that strong convergence rate with respect to the parabolic grid meshsize is expected to be optimal. We show that one can reach almost sure convergence rate (and no better) when measuring the error in appropriate negative Besov norms, by temporarily `pretending' that the SPDE is singular.
Paper Structure (15 sections, 22 theorems, 53 equations, 3 figures)

This paper contains 15 sections, 22 theorems, 53 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The setup
  3. Function spaces
  4. Main result
  5. Various tools
  6. Kernels and convolutions
  7. Properties of discrete and continuous function spaces
  8. Heat kernel bounds
  9. Heat kernel error bounds
  10. Main proofs
  11. Bounds on the linear solutions in Besov norms
  12. A priori bounds on the approximations
  13. Proof of Theorem \ref{['thm:main']}
  14. Implementation
  15. Some technical proofs

Key Result

Theorem 1.7

Let $\theta\in(-1/2,0]$ and $\varepsilon\in(0,1/2+\theta)$. Assume that $\psi\in C^{1-\varepsilon}(\mathbb{T})$. Then there exists an almost surely finite random variable $\eta$ such that for all $n\in\mathbb{N}$

Figures (3)

  • Figure 1: A log-log plot of the error (compared to the highest resolution approximation) for a single realisation of the approximations, measured in $L_\infty$, $L_1$, and tested against a single test function, showing superior rate for the latter.
  • Figure 2: A fine grid approximation at time $1$ (left), two coarser grid approximations (middle) and their differences (right). The error plots show oscillatory behaviour around $0$, which suggests smaller error in negative regularity spaces, agreeing with Theorem \ref{['thm:main']}.
  • Figure 3: A fine grid approximation (left), a coarse grid approximation (middle) and their difference (right) for the Euler scheme applied to the SDE $dX_t=2X_t\,dt+(\arctan(X_t)-\pi/2)\,dW_t$. Here the error is one-sided and thus not expected to be smaller in distributional norm, as discussed in Remark \ref{['rem:SDE']}.

Theorems & Definitions (50)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 40 more